Regression with Respect to Sensing Actions and Partial States

In this paper, we present a state-based regression function for planning domains where an agent does not have complete information and may have sensing actions. We consider binary domains , and employ the 0-approximation (Son & Baral 2001) to define the regression function. In binary domains, the use of 0-approximation means using 3-valued states. Although planning using this approach is incomplete, we adopt it to have a lower complexity. We prove the soundness and completeness of our regression formulation with respect to the definition of progression and develop a conditional planner that utilizes our regression function. Introduction and Motivation An important aspect in reasoning about actions and in characterizing the semantics of action description languages is to define a transition function encoding the transition between states due to actions. This transition function is often viewed as a progression function in that it denotes the progression of the world by the execution of actions. The ‘opposite’ or ‘inverse’ of progression is referred to as regression. Even for the simple case where we have only non-sensing actions and the progression transition function is deterministic, there are various formulations of regression. For example, consider the following. Let Φ be the progression transition function from actions and states to states. I.e., intuitively Φ(a, s) = s′ means that if the action a is executed in state s then the resulting state will be s′. One way to define a regression function Ψ1 is to define it with respect to states. In that case s ∈ Ψ1(a, s′) will mean that the state s′ is reached if a is executed in s. Another way regression is defined is with respect to formulas. In that case Ψ2(a, f) = g, where f and g are formulas, means that if a is executed in a state satisfying g then a state satisfying f will be reached. For planning using heuristic search often a different formulation of regression is given. Since most planning research is about goals that are conjunction of literals, regression is defined with respect to a set of literals and an action. In that case the conjunction of literals (often specifyCopyright c © 2004, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. Due to space limitation, we do not present the results for nonbinary domains here. Interested readers can find such results at http://www.public.asu.edu/ ̃lctuan/TOCL/ ing the goal) denotes a set of states, one of which needs to be reached. This regression is slightly different from Ψ2 as the intention is to regress to another set of literals (not an arbitrary formula), denoting a sub-goal. With respect to the planning language STRIPS, where each action a has an add list Add(a), a delete list Del(a), and a precondition list Prec(a), the progression function is defined as Progress(s, a) = s + Add(a) − Del(a); and the regression function is defined as Regress(conj, a) = conj+Prec(a)−Add(a), where conj is a set of atoms. The relation between these two, formally proven in (Pednault 1986), shows the correctness of regression based planners; which in recent years through use of heuristics (e.g. (Bonet & Geffner 2001; Nguyen, Kambhampati, & Nigenda 2002)) have done exceedingly well on planning competitions. In this paper we are concerned with domains where the agent does not have complete information about the world, and may have sensing actions, which when executed do not change the world, but rather give certain information about the world to the agent. As a result, plans may now no longer be simply a sequence of (non-sensing) actions but may include sensing actions and conditionals. Various formalisms have been developed for such cases (e.g. (Lobo 1998; Son & Baral 2001)) and progression functions have been defined. Also, the complexity of planning in such cases has been analyzed in (Baral, Kreinovich, & Trejo 2000). One approach to planning in the presence of incomplete information is conformant planning where no sensing action is used, and a plan is a sequence of actions leading to the goal from every possible initial situation. However, this approach proves inadequate for many planning problems (Son & Baral 2001), i.e., there are situations where sensing actions are necessary. In that case, one approach is to use belief states or Kripke models instead of states. It is shown that the total number of belief states is double exponential while the total number of 3-valued states is exponential in the number of fluents (Baral, Kreinovich, & Trejo 2000). Here, we pursue a provably less complex formulation with sensing actions and use 3-valued states. In this approach, we will miss certain plans, but that is the price we are willing to pay for reduced complexity. This is consistent with and similar to the considerations behind conformant planning. With that tradeoff in mind, in this paper we consider the 0-approximation semantics defined in (Son & Baral 2001) and define regression with respect to that semantics. We then formally relate our definition of regression with the earlier definition of progression in (Son & Baral 2001) and show that planning using our regression function will indeed give us correct plans. We then use our regression function in planning with sensing actions and show that, even without using any heuristics, our planner produces good results. To simplify our formulation, we only consider STRIPS like actions where no conditional effects are allowed. We also restrict domain of a fluent to a finite set of discrete values. In summary the main contributions of our paper are: • A state-based regression function corresponding to the 0approximation semantics in (Son & Baral 2001); • A formal result relating the regression function with the progression transition function in (Son & Baral 2001); • An algorithm that uses these regression functions to construct conditional plans with sensing actions; • Implementation of this algorithm; and • Illustration of the performance of this algorithm with respect to several examples in the literature. Related Work Our work in this paper is related to different approaches to regression and planning in the presence of sensing actions and incomplete information. It differs from earlier formula regression such as (Pednault 1994; Reiter 2001; Son & Baral 2001) in that it is a state-based formulation and the other are formula based. Unlike the conditional planners (Peot & Smith 1992; Cimatti, Roveri, & Traverso 1998), our planner can deal with sensing actions similar to the planners in (Etzioni et al. 1992; Lobo 1998; Son, Tu, & Baral 2004; Weld, Anderson, & Smith 1998). However, it does not deal with nondeterministic and probabilistic actions such as the planners in (Bonet & Geffner 2001; Pryor & Collins 1996; Rintanen 2000; 2002). It is also not a conformant planner as in (Cimatti, Roveri, & Traverso 1998; Eiter et al. 2000). For these reasons, we currently compare our planner with those of (Son, Tu, & Baral 2004; Weld, Anderson, & Smith 1998). Background: 0-Approximation Semantics For A STRIPS-like Language Action and Plan Representation We employ a STRIPS-like action representation (Fikes & Nilson 1971) and represent a planning problem by a tuple P = 〈A, O, I,G〉 where A is a finite set of fluents, O is a finite set of actions, and I and G encode an initial state and a goal state, respectively. A fluent literal is either a positive fluent f ∈ A or its negation (negative fluent) ¬f . In this paper, we are interested in the planning problem in which I and G are sets of fluent literals. An action a ∈ O is either a non-sensing action or a sensing action and is specified as follows: • A non-sensing action a is specified by an expression of the form action a :Pre Prea :Add Adda :Del Dela where Prea is a set of fluent literals representing the precondition for a’s execution, Adda and Dela are two disjoint sets of positive fluents representing the positive and negative effects of a, respectively; and • A sensing action a is specified by an expression of the form action a :Pre Prea :Sense Sensa where Prea is a set of fluent literals and Sensa is a set of positive fluents that do not appear in Prea. To illustrate the action representation and our search algorithm, we will use a small example, a version of the “Getting to Evanston” from (Weld, Anderson, & Smith 1998). Figure (1) shows the actions of this domain. Non-sensing action :Pre :Add :Del goto-western-at-belmont at-start on-western at-start